The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by

# The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by

H
240 points

The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by

$$f(n) = \begin{cases} ax^2 e^{-bx^2}, & \text{x \ge 0} \\ 0, & \text{x < 0} \end{cases}$$

where $b = m/2kT$ and $k$, $T$, and $m$ denote, respectively, Boltzmann’s constant, the absolute temperature of the gas, and the mass of the molecule. Evaluate a in terms of b.

probability density
molecule speed
random variable

426.1k points

the function f is a probability density function, so it must integrate to 1. To solve this integral use integration by parts:

$$\int_0^{\infty}ax^2 e^{-bx^2}=-\frac{axe^{-bx^2}}{b} \, \bigg|_0^{\infty}+\frac{a}{b}\int_0^{\infty}e^{-bx^2}$$

$$=0-0+\frac{a}{b}\int_0^{\infty}e^{-bx^2}$$

theres a trick to solving the second integral here

$$=\frac{a}{b}\frac{\sqrt{\pi}}{2\sqrt{b}}=1$$

solve this for a in terms of b giving,

$$a=\frac{2b^{3/2}}{\sqrt{\pi}}$$

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